E2-formality conjecture for the quantum-group endomorphism scheme

Let $\mathscr{X}$ be the $E_3$-scheme over the flag variety associated to the derived endomorphism algebra of the unit for the half-quantum flag variety. Let $\widetilde{\mathcal{N}}$ be the Springer resolution, and let $\operatorname{QCoh}_{\mathrm{dg}}$ denote dg quasi-coherent sheaves.

Formality conjecture. The $E_3$-scheme $\mathscr{X}$ is $E_2$-formal, so that there is a monoidal equivalence

$$\operatorname{QCoh}_{\mathrm{dg}}(\widetilde{\mathcal{N}})\overset{\sim}\to \operatorname{QCoh}_{\mathrm{dg}}(\mathscr{X}).$$

Consequently, $\operatorname{QCoh}_{\mathrm{dg}}(\widecheck{G}/B_q)$ has the structure of a sheaf of tensor categories over a twisted version of the Springer resolution.

This conjecture would supply the monoidal enhancement needed to view the quantum-group category as a sheaf of tensor categories over the Springer resolution. The source does not give evidence that it has been proved.